Full waveform inversion (FWI) has received an increasing amount of attention thanks to its ability to provide a high resolution velocity model of the subsurface. The computational cost still presents a challenge, however, and the convergence rate of the FWI problem is usually very slow without proper preconditioning on the gradient. While preconditioners based on the Gauss-Newton Hessian matrix can provide significant improvements in the convergence of FWI, computation of the Hessian matrix itself has been considered highly impractical due to its computational time and the storage requirements. In this paper, we design preconditioners based on an approximate Gauss-Newton Hessian matrix obtained using the phase-encoding method. The new method requires only 2Ns forward simulations compared to Ns(Nr+1)forward simulations required in conventional approaches, where Ns and Nr are the numbers of sources and receivers. We apply the diagonal of the phase-encoded Gauss-Newton Hessian to both sequential source FWI and encoded simultaneous source FWI. Numerical examples on Marmousi model demonstrate that phase-encoded Gauss-Newton Hessian improves the convergence of the FWI significantly.


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